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Geometry and Physics of Hitchin Systems: January 22 – June 21, 2019

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Organized by Lara Anderson and Laura Schaposnik

Hitchin systems have found remarkable applications in many different areas of mathematics and physics. In particular, Hausel and Thaddeus related Higgs bundles to mirror symmetry, and in the work of Kapustin and Witten, Higgs bundles were used to give a physical derivation of the geometric Langlands correspondence. Moreover, in the last year many advances were made in the study of branes inside the moduli space of Higgs bundles, opening paths to promising new research directions.

The overall goal of the program is to bring together mathematicians and physicists working on areas close to Higgs bundles: the four interrelated themes of the program are: (A) Higgs bundles and geometric structures; (B) Higgs bundles and representation theory; (C) Higgs bundles and mirror symmetry; (D) Higgs bundles and singular geometry. Both (A) and (B) are closely interrelated, and concern in particular the study of the geometry of the moduli space, and the understanding of limits and geometric structures on this space. In particular, since many geometric structures are parametrised by the Hitchin fibration, and limits are taken in this context too, we hope that the workshop will lead to the understanding of these questions in the context of (B) and (C). Indeed, physicists have long been interested in the appearance of the Coulomb Branch (an Abelianized limit of an associated physical gauge theory), the Hitchin base, and Seiberg-Witten curves, the spectral covers of Higgs bundles. Finally, there should be algebraic connections between the above limits and Donaldson-Thomas type of invariants and wall-crossings, either in the Hitchin base or in the total space, and hence algebraic geometers can develop useful tools from this perspective. Among possible directions related to (C) is the study of the moduli space of Higgs bundles and its branes through the Hitchin fibration, appearing in mathematical physics and Langlands duality. It is from this perspective that mathematical physicists and theoretical physicists will play a key role in bringing attention to the most urgent problems in the area, and also for developing new methods centred around spectral data of Higgs bundles. Moreover, since one can use equivariant characters of certain coherent sheaves on the Hitchin moduli space to further understand the physics proposal of dualities between branes, interactions with the algebraic geometers will be of significant importance during the workshop. February 5, 2018. 1 2 Finally, (D), which links Calabi-Yau and Hitchin Integrable systems, is another rapidly developing area that we hope to highlight. Recent work has brought to light a remarkable isomorphism between (non-compact) Calabi-Yau integrable systems and those of Hitchin. Extensions of this correspondence to compact Calabi-Yau geometry have recently been found in the context of string compactifications but many open questions remain. In addition, links between Hitchin and Calabi-Yau moduli spaces have the potential to shed light on a number of important physical questions regarding string vacuum spaces. We expect the program to have a strong broader impact, emphasizing the presence of women and other minorities working in the area, and promoting the area amongst younger researchers.

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Program Application Deadline: October 23, 2019 (or when event is at maximum capacity). Applicants will be contacted soon after this date